Advertisement

Computing and Visualization in Science

, Volume 16, Issue 2, pp 59–76 | Cite as

Adaptive AMG with coarsening based on compatible weighted matching

  • Pasqua D’Ambra
  • Panayot S. Vassilevski
Article

Abstract

We introduce a new composite adaptive Algebraic Multigrid (composite \(\alpha \)AMG) method to solve systems of linear equations without a-priori knowledge or assumption on characteristics of near-null components of the AMG preconditioned problem referred to as algebraic smoothness. Our version of \(\alpha \)AMG is a composite solver built through a bootstrap strategy aimed to obtain a desired convergence rate. The coarsening process employed to build each new solver component relies on a pairwise aggregation scheme based on weighted matching in a graph, successfully exploited for reordering algorithms in sparse direct methods to enhance diagonal dominance, and compatible relaxation. The proposed compatible matching process replaces the commonly used characterization of strength of connection in both the coarse space selection and in the interpolation scheme. The goal is to design a method leading to scalable AMG for a wide class of problems that go beyond the standard elliptic Partial Differential Equations (PDEs). In the present work, we introduce the method and demonstrate its potential when applied to symmetric positive definite linear systems arising from finite element discretization of highly anisotropic elliptic PDEs on structured and unstructured meshes. We also report on some preliminary tests for 2D and 3D elasticity problems as well as on problems from the University of Florida Sparse Matrix Collection.

Keywords

Adaptive AMG Weighted matching Strength of connection Compatible relaxation 

Mathematics Subject Classification

65F10 65N55 

Notes

Acknowledgments

We wish to thank Bora Uçar for stimulating discussions and his advice on available algorithms and software for computing weighted matching in bipartite and general graphs. We also thank one of the authors of MFEM, Tzanio Kolev for his kind help in installing and using MFEM. Finally, we thank one of the reviewers for motivating us to run more examples and use one more version of coarsening that demonstrated better the potential of the proposed composite \(\alpha \)AMG solver.

References

  1. 1.
    Baker, A.H., Falgout, R.D., Kolev, T.V., Yang, U.M.: Multigrid smoothers for ultra-parallel computing. SIAM J. Sci. Comput. 33, 2864–2887 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brandt, A.: General highly accurate algebraic coarsening. Elect. Trans. Num. Anal. 10, 1–20 (2000)MATHGoogle Scholar
  3. 3.
    Brandt, A.: Multiscale scientific computation: review 2001. In: Barth, T.J., Chan, T.F., Haimes, R. (eds.) Multiscale and Multiresolution Methods: Theory and Applications, pp. 1–96. Springer, Heidelberg (2001)Google Scholar
  4. 4.
    Brandt, A., Brannick, J., Kahl, K., Livshitz, I.: Bootstrap AMG. SIAM J. Sci. Comput. 33, 612–632 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brandt, A., McCormick, S., Ruge, J.: Algebraic multigrid (AMG) for sparse matrix equations. In: D. J. Evans (ed.) Sparsity and its Applications (1984)Google Scholar
  6. 6.
    Brannick, J., Falgout, R.D.: Compatible relaxation and coarsening in algebraic multigrid. SIAM J. Sci. Comput. 32, 1393–1416 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brannick, J., Chen, Y., Kraus, J., Zikatanov, L.: Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs. SIAM J. Num. Anal. 51, 1805–1827 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brezina, M., Falgout, R.D., MacLachlan, S., Manteuffel, T., McCormick, S., Ruge, J.: Adaptive smoothed aggregation \(\alpha \) SA multigrid. SIAM Rev. 47, 317–346 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Brezina, M., Falgout, R.D., MacLachlan, S., Manteuffel, T., McCormick, S., Ruge, J.: Adaptive algebraic multigrid. SIAM J. Sci. Comput. 27, 1261–1286 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diestel R.: Graph Theory. Springer, Heidelberg, GTM 173, 4th ed. (2010)Google Scholar
  11. 11.
    Drake, D.E., Hougardy, S.: A linear time approximation algorithm for weighted matchings in graphs. ACM Trans. Algorit. 11, 107–122 (2005)Google Scholar
  12. 12.
    Duff, I.S., Koster, J.: On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Anal. Appl. 22, 973–996 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Duff, I.S., Uçar, B.: Combinatorial Problems in Solving Linear Systems, Rutherford Appleton Laboratory Technical Report RAL-TR-2008-014 and TR/PA/08/26, CERFACS, ToulouseGoogle Scholar
  14. 14.
    Falgout, R.D., Vassilevski, P.S.: On generalizing the AMG framework. SIAM J. Num. Anal. 42, 1669–1693 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Halappanavar, M., Feo, J., Villa, O., Tumeo, A., Pothen, A.: Approximate weighted matching on emerging manycore and multithreaded architectures. Int. J. High Perform. Comput. Appl. first published on August 9, 2012 as doi: 10.1177/1094342012452893
  16. 16.
    HSL(2011). A Collection of Fortran Codes for Large Scale Scientific Computation. http://www.hsl.rl.ac.uk
  17. 17.
    hypre: High Performance Preconditioners. http://www.llnl.gov/CASC/hypre/
  18. 18.
    Karypis, G.: METIS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices, Ver. 5.1.0, University of Minnesota, (2013)Google Scholar
  19. 19.
    Kolev, T. V., Vassilevski, P. S.: Parallel Auxiliary Space AMG for H(curl) problem. J. Comput. Math. 27, 604–623 (2009). Special issue on Adaptive and Multilevel Methods in Electromagnetics. UCRL-JRNL-237306Google Scholar
  20. 20.
    Lashuk, I., Vassilevski, P.S.: On some versions of the element agglomeration AMGe method. Num. Linear Alg. Appl. 15, 595–620 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Livne, O.E.: Coarsening by compatible relaxation. Num. Linear Alg. Appl. 11, 205–227 (2004)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Manne, F., Bisseling, R.H.: A Parallel Approximation Algorithm for the Weighted Maximum Matching Problem, in PPAM 2007, LNCS, vol. 4967, pp. 708–717. Springer, Berlin (2008)Google Scholar
  23. 23.
    Nägel, A., Falgout, R.D., Wittum, G.: Filtering algebraic multigrid and adaptive strategies. Comput. Vis. Sci. 11, 150–167 (2008)CrossRefGoogle Scholar
  24. 24.
    Notay, Y.: An aggregation-based algebraic multigrid method. Elect. Trans. Num. Anal. 37, 123–146 (2010)MathSciNetMATHGoogle Scholar
  25. 25.
    Napov, A., Notay, Y.: An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34, A1079–A1109 (2012)Google Scholar
  26. 26.
    Olschowka, M., Neumaier, A.: A new pivoting strategy for Gaussian elimination. Linear Alg. Appl. 240, 131–151 (1996)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Parter, S.: The use of linear graphs in Gaussian elimination. SIAM Rev. 3, 119–130 (1961)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Preis, R.: Linear Time 1/2-Approximation Algorithm for Maximum Weighted Matching in General Graphs, in STACS’99. LNCS vol. 1563, pp 259–269. Springer, Berlin (1999)Google Scholar
  29. 29.
    Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56, 179–196 (1996)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Vassilevski, P.S.: Multilevel Block Factorization Preconditioners, Matrix-based Analysis and Algorithms for Solving Finite Element Equations. Springer, New York, NY (2008)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute for High-Performance Computing and NetworkingNational Research Council of ItalyNaplesItaly
  2. 2.Center for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermoreUSA

Personalised recommendations